Area of Isosceles Triangle - (Formulas, Derivation and Examples)

An isosceles triangle is defined as a type of triangle which has any two sides equal in length. The two angles of an isosceles triangle are opposite to equal sides and are equal in measure. In geometry, the triangle is a three-sided polygon which is classified into three categories based on its sides, they are given below-

• Scalene triangle (In a scalene triangle all three sides are unequal)

• Isosceles triangle (In an isosceles triangle only two sides are equal)

• Equilateral triangle (In an equilateral triangle all three sides are equal)

What Is The Isosceles Triangle?

An Isosceles triangle is a triangle that always has two equal sides and the two angles opposite the two equal sides are equal.

Suppose in a \triangle\:ABC , if sides AB and AC are equal, then \triangle\:ABC is an isosceles triangle where \angle\:B=\angle\:C So this theorem describes that the isosceles triangle is “if the two sides of a triangle are equal, then the angle opposite to them is always equal”.

Angles Of Isosceles Triangle

The two of the three angles of the isosceles triangle are always equal in measure, which is opposite to the equal sides. Therefore, one of the angles is unequal.

Properties Of Isosceles Triangle

1. As the two sides are equal in an isosceles triangle then the unequal side is called the base of the triangle

2. The angles opposite to the two equal sides of the isosceles triangle are always equal

3. The altitude of an isosceles triangle is measured from the base to the vertex or from the topmost of the triangle

4. A right isosceles triangle has a third angle which is ‘90’ degrees

Isosceles Triangle Theorem

As per the isosceles triangle theorem, if any two sides are congruent in an isosceles triangle, then the angles opposite to the two sides are also congruent.

Similarly, if any two angles are congruent in an isosceles triangle, then the sides opposite to them are also congruent.

In the given triangle ABC,

AB = AC

\angle\:ABC=\angle\:ACD

Types Of Isosceles Triangle

The isosceles triangle is classified into three types, which are given below-

1.Isosceles Acute Triangle

As we know, the different dimensions of a triangle are its legs, base, and height. All the isosceles triangles have an axis of symmetry with the perpendicular bisector of their base. Depending on the angle between the two legs, the isosceles triangle is classified into acute, right and obtuse. The isosceles triangle can be acute if any two angles opposite the legs are equal and all angles are less than 90 degrees which are said to be acute angles.

2. Isosceles Right Triangle

A right isosceles triangle always has two equal sides, one of the two equal sides acts as perpendicular and another one acts as a base of the triangle and the third side, which is unequal, is called the hypotenuse. We can apply here Pythagoras, which states that the square of the hypotenuse is equal to the sum of the square of the base and perpendicular.

Let us Suppose the sides of the right isosceles triangle are a, a, and h

h=\left(\sqrt{a^2+a^2}\right)=\left(\sqrt{a^2}\right)=a \sqrt{2} \text { or } h=\sqrt{2} a

here a is the two equal sides, and h is the hypotenuse.

3. Isosceles Obtuse Triangle

An obtuse triangle is defined as a triangle in which one of its angles is greater than 90 degrees which are said to be a right angle. The obtuse triangle can be a scalene triangle or an isosceles triangle. Therefore, the isosceles obtuse triangle is a triangle which always has two equal sides with an obtuse angle.

Isosceles Triangle Formulas

We know that an isosceles triangle is a two-dimensional or 2-D in shape with three sides. The measures of an isosceles triangle are the area and perimeter.

Area Of Isosceles Triangle

The area of an isosceles triangle is the region which is occupied by it in the two-dimensional or 2-D space. The area of an isosceles triangle is also defined as half the product of the base and height of an isosceles triangle. The formula to calculate the area of an isosceles triangle is given below-

The area of an isosceles triangleA=\frac{1}{2}\times\:b\times\:h

Square units

here b is equal to the base and h is equal to the triangle’s height.

The Perimeter Of The Isosceles Triangle

Any shape's border is referred to as the shape's perimeter. Similarly, the perimeter of an isosceles triangle is defined as the sum of any three sides of an isosceles triangle. If we know its base and side then the perimeter of an isosceles triangle can be found. The formula to calculate the perimeter of the isosceles triangle is given below-

The perimeter of an isosceles Triangle, P=(2a+b)units

Here ‘a’ is equal to the length of the two equal legs of an isosceles triangle and ‘b’ is equal to the base of the triangle.

Isosceles Triangle Altitude

When an altitude is drawn to the base of the isosceles triangle then it always bisects the vertex angle and if it bisects the base then the two congruent triangles are created. The altitude of the triangle forms the required right angle and the altitude which becomes the shared legs between the created two congruent triangles. The congruent legs of a triangle become the congruent hypotenuse and the altitude drawn to the base of the isosceles triangle bisects the base.